center for electronic correlations and magnetism university of … · 2019-11-12 · • dynamical...

Dieter Vollhardt

The art of modeling in solid state physics

Johannes Kepler Universität Linz; November 6, 2019

Center for Electronic Correlations and MagnetismUniversity of Augsburg

• Modeling in the natural sciences

• Step 1: From materials to models

• Electronic correlations

• Dynamical mean-field theory

• Step 2: From models back to materials

Outline

Modeling in the natural sciences

Observation Interpretation Modeling→ ↔

Micro-cosmos Macro-cosmos

Observation and interpretation

• What is observed?• How to interpret it?• How can it be explained?

Fundamental research

Requires idealization and abstraction (“reductionism“) models↔

The art of modeling

Weighing scale

Model (Archimedes)

Every-day use

©ge

ttyi

mag

es

Spherical lodestone: model of Earth

⇒ Earth is a magnet

1600 William GILBERT of Colchester

De Magnete

(“Father of magnetism“)

1269 Pierre PELERIN de Maricourt: spherical lodestone (=magnet) has poles

The art of modeling

Earth‘s magnetic field

©Sh

utte

rsto

ck

Known since antiquity

Papin (1680)

blac

inc.

com

The art of modeling

Steam engine

Every-day use

Simple models

Mechanical models: very successful

Electromagnetism

The art of modeling

vortices

Allowed to discuss the displacement current, but… Maxwell equations: 1864

J. C. Maxwell (1857):"I have been grinding at … a vortical theory of magnetism and electricity which is very crude but has some merits."

electrical current of particles rotation of vortices gh opposite motion of particles above gh

(induced opposite electric current) anti-rotation transmitted to vortices kl, …

layers of particles(“rolling contact”)

“It seems to me that the test of ‘Do we or do we not understand a particular point in physics ?’ is, ‘Can we make a mechanical model of it ?’ “ Lord Kelvin (1884)

Most models are not successful

Maxwell’s mechanical model of electromagnetism: Vortices in a molecular medium

The art of modeling

Bohr model of the atom Bohr (1913)

Circumference de Broglienu nλ=

+ Postulates:• discrete energies• frequency of emitted light • quantization condition for orbits

/n mE E h= −

• Correspondence principle• Semiclassical limit of quantum mechanics

Example: Iron (Fe)

Microscopic view: O(1023) interacting electrons + ions

The art of modeling

Properties of solids

Macroscopic view:

How to investigate/model?

M(T)

T TC

How to understand the• origin of ferromagnetism in Fe• T-dependence of the magnetization M ?

Simplest quantum model for interacting electrons in a solid:

modeling

Þ

H

eld, 2004

Theoretical investigation of materials

Fe: Interacting many-particle system Maximal simplification needed(… more art than science)

Hubbard model (1963)to explain ferromagnetism in transition metals

Step 1: From materials to models

How to explain ferromagnetism in transition metals?

skul

lsin

thes

tars

.com

Weiss model of magnetic domains (1906)

Alignment of “elementary magnets” in each domain due to a “molecular field“

(Weiss mean field)

Microscopic origin?


Z. Physik 31, 253 (1925)

web

.sta

nfor

d.ed

u

Exact solution in d=1: no order (also in d=3?!)

Ising(-Lenz) model (1925)

But: Magnetism is a quantum effect Bohr (1911), van Leeuwen (1919)


Classical spin model

Z. Physik 49, 619 (1928)

But: Electrons are mobile Bloch (1929)

Heisenberg model (1928)

Weiss molecular field originates fromquantum-mechanical exchange processes

com

plex

ity-

cove

ntry

.org

must include their kinetic energy + Coulomb interaction


Quantum spin model

Next year: Hubbard model ??

Not for another 34 years!

1929

Development of many-body theory for condensed matter:

• Field-theoretic/diagrammatic methods (Feynman 1949)

• Superconductivity (Bardeen, Cooper, Schrieffer 1957)

• Single-impurity model (Anderson 1961)

• Elementary excitations/quasiparticles, Fermi liquid theory (“Standard model of condensed matter physics“) (Landau 1956)

U

Non-interactingconduction (s-) electrons

+Immobile d-electrons with

interaction U on a single site ("impurity")

Single-impurity Anderson model Anderson (1961)

t

Us,d-hybridization V

+

• Characteristic 3-peak structure• non-perturbative energy scale("Kondo physics")

Bulla, Hewson, Pruschke (1998)

Single-impurity Anderson model Anderson (1961)

Non-interactingconduction (s-) electrons

+Immobile d-electrons with

interaction U on a single site ("impurity")

t

Two fundamental unsolved problems

Beginning of the 1960s

Ni,V 3d electrons form narrow bands (“correlated”): How to model?

Tem

pera

ture

(K)

Pressure (4 kbar/division)

Electronic Correlations

AB A B≠

quantifies correlations

AB A B−

Correlations are present when

Wigner (1934)

Definition of electronic correlations (I): Effects beyond factorization of the interaction (Hartree-Fock)

e.g., 2( ) ( ') ( ) ( ')n n n n n≠ =r r r r

Narrow band systems :

Interaction correlations magnetismkin intW E E<

Pauli principle

Unusual properties of correlated electron materials

• huge resistivity changes• gigantic volume anomalies• colossal magnetoresistance

• high-Tc superconductivity

• correlated metallic behavior at interfaces of insulators, …

• sensors, switches• thermoelectrics• high-Tc superconductor devices• functional materials: oxide heterostructures, …

Interesting for technological applications:

• How to explain these properties?• What is the minimal model?

Diagonal in momentum space(waves)

Diagonal in position space(particles)

,

, UH n D D n nσσ

ε ↑ ↓= + =∑ ∑k i i i ik i

k

Gutzwiller (1963)Hubbard (1963)Kanamori (1963)• tight binding

• extreme screening: local interaction allowedby Pauli principle

no classical analogue

Hubbard model

Minimal lattice model for correlated electrons

• How to solve?• No fully numerical solutions possible• Find good approximation (mean-field?)

Single-band model:

1. Construction by factorization

- Spin models (finite range interaction), e.g., IsingFactorization Weiss MFT

Mean-Field Theory (MFT)

- Spin S- Degeneracy N- Range of interaction R - Spatial dimension d or coordination number Z

}2. Construction by exaggeration

→∞

Hypercubic lattices: Z=2d

- Electronic models (local interaction), e.g., HubbardFactorization Hartree MFT

Ising model: all yield Weiss MFT

n n n n↑ ↓ ↑ ↓→i i i i

S S S S→i j i j

Mean-field theory of the Hubbard model

( )ωΣ

Metzner, DV (1989)

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Purely local interaction:independent of d,Z

,d Z→∞→ • what simplifications arise ?• what type of MFT ?

n n n n↑ ↓ ↑ ↓≠i i i i

Quantum fluctuations neglected static (no correlations)

Local quantum fluctuations always present dynamic

No factorization: very different from Hartree-Fock


( )ωΣMüller-Hartmann (1989)

Janiš (1991)

Metzner, DV (1989)

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Strong diagrammaticsimplifications


mutuallydependent

Self-consistent mean-field theory

Georges, Kotliar (1992), Jarrell (1992)

( )ωΣ


Janiš (1991)Self-consistent mean-field theory

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Metzner, DV (1989)


Dynamical mean-field theory (DMFT)

Hubbard model single-impurity Anderson model+ self-consistency condition

→∞→,d Z

Strong diagrammaticsimplifications

DMFT:

• Fully dynamical, but mean-field in position space

• New type of MFT for quantum particles

DMFT of the Hubbard model: Summary

,d Z →∞• Exact in

( , )ωΣ k ( )ωΣKotliar, DV (2004)

Characteristic features of DMFT

U

EEd+U/2Ed-U/2

U

Ed

Atomic limit (no kinetic energy)

( , )ωΣ k ( )ωΣ

Density of states/Spectral function

Kotliar, DV (2004)

EEd+U/2Ed-U/2

U

Ed

lowerHubbard bandincoherent

upperHubbard bandincoherent

Excitations at Fermi level

(quasiparticles)

fermions (with kinetic energy)

U( , )ωΣ k ( )ωΣ

Kotliar, DV (2004)



coherent

U

Experimentallydetectable

(PES, ARPES, …)

Definition of electronic correlations (II):• transfer of spectral weight• finite lifetime of excitations

U( , )ωΣ k ( )ωΣ

Kotliar, DV (2004)



Application of DMFTMott metal-insulator transition

Intermediate-coupling problemin

sula

tor

met

al

lowerHubbard band

upperHubbard band

quasiparticles

Mott metal-insulator transition

→

ParamagneticMott insulator

P←Kotliar, DV (2004)

Mott metal-insulator transition: phase diagram

V2O3κ-organics, ...

V2O3

P →McWhan, Menth, Remeika, Brinkman, Rice (1973)


Fermi liquid

Critical point

P →Kotliar, DV (2004)


U←


Pustogow, Rösslhuber, Tan, Uykur, Böhme, Löhle, Hübner, Schlueter, Dressel, Dobrosavljević(arXiv:1907.04437)

U


Fermi liquid

Critical point

P →Kotliar, DV (2004)


←


Step 2: From models back to materialsApplication of DMFT to correlated materials

Combination ?

, GGA

Non-perturbative approaches for real materials

UHel

d (2

004)

Hel

d (2

004)

+Local electronic correlations

Double counting correction

(Many-body theory: DMFT)

LDA+DMFTAnisimov, Poteryaev, Korotin, Anokhin, Kotliar (1997)Lichtenstein, Katsnelson (1998)

Material specific electronic structure(Density functional theory: LDA)

Computational scheme for correlated electron materials

=

-

Simplest approach:

Early application of DFT+DMFT

(Sr,Ca)VO3: 3d1 system

TheoryElectronic structure

180V VO∠ − − = °

162V VO∠ − − ≈ °

No correlation effects/spectral transfer

LDA+DMFT results

Constrained LDA:U=5.55 eV, J=1.0 eV

k-in

tegr

ated

spe

ctra

l fun

ctio

n A

(ω) , CaVO3

Osaka – Augsburg – Ekaterinburg collaboration:Sekiyama et al. (2004)

Pavarini et al. (2004)

Spectral function ( , )A ωkSrVO3:

Byczuk et al. (2007)

LDA

LDA+DMFT

Electronic correlations • band narrowing• quasiparticle damping

occupied states unoccupied states

LDA+DMFT for (Sr,Ca)VO3: Comparison with experiment

Correlation-induced3-peak structure

confirmed

Sekiyama et al. (2004, 2005) [Osaka – Augsburg – Ekaterinburg collaboration]

One-band Hubbard model on a generalized fcc lattice ( )Z →∞

Paradox “spins vs. electrons” reconciled

Curie-WeissBrillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #

Intermediate-coupling problem

TC

Application of DMFTMetallic ferromagnetism

mean-field critical exponents

( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −

1 , 12

β γ= =

Ulmke (1998)

Ulmke (1998)

Curie-Weiss


( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −

LDA+DMFT

Microscopic origin of exchange interactions in Fe

Fe,Ni

TC

Lichtenstein, Katsnelson, Kotliar (2001)

Kvashnin et al. (2016)


Brillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #

1 , 12

β γ= =

0.35, 1.33β γ≈ ≈Exp.:


Ulmke (1998)

Curie-Weiss


( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −

LDA+DMFT Fe at Earth’s core conditions

TC

Hausoel et al. (2017)Pourovskii (2019)


Brillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #

1 , 12

β γ= =

astr

obob

.are

avoi

ces.

com


- transition-metal oxides- f-electron systems- cuprates, iron pnictides, …

Jacob, Haule, Kotliar (2009,2010)al-Badri et al. (arXiv:1811.05739)

Molecular electronics, quantum chemistry, ligand binding

Bulk materials

Heterostructures, interlayers, surfaces

Assm

ann

et a

l.(2

013) Okamoto, Millis (2004)

Janson, Held (2018)

Ni dimer

Cu nanowire Cu nanowire

Lechermann (2018)

DMFT: Applications from bulk matter to heterostructures

Lech

erm

ann

(201

8)

ca. 150 lecture notes freely available on:https://www.cond-mat.de/events/correl.html

DMFT: Generic mean-field theory of correlated electrons Wide field of applications

http://gamediv1.weebly.comHemberger et al. (2002)

Bruker: EPR in Life SciencePauli, Willmott (2008)

±∞

Perfetti et al. (2006)

Pustogow et al, arXiv:1907.04437

center for electronic correlations and magnetism university of … · 2019-11-12 · • dynamical...

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